Stability of Twisted States in the Kuramoto Model on Cayley and Random Graphs

The Kuramoto model (KM) of coupled phase oscillators on complete, Paley, and Erdős-Rényi (ER) graphs is analyzed in this work. As quasirandom graphs, the complete, Paley, and ER graphs share many structural properties. For instance, they exhibit the same asymptotics of the edge distributions, homomorphism densities, graph spectra, and have constant graph limits. Nonetheless, we show that the asymptotic behavior of solutions in the KM on these graphs can be qualitatively different. Specifically, we identify twisted states, steady state solutions of the KM on complete and Paley graphs, which are stable for one family of graphs but not for the other. On the other hand, we show that the solutions of the IVPs for the KM on complete and random graphs remain close on finite time intervals, provided they start from close initial conditions and the graphs are sufficiently large. Therefore, the results of this paper elucidate the relation between the network structure and dynamics in coupled nonlinear dynamical systems. Furthermore, we present new results on synchronization and stability of twisted states for the KM on Caley and random graphs.